A thermo-hydro-mechanical constitutive model and its numerical modelling for unsaturated soils

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A thermo-hydro-mechanical constitutive model

and its numerical modelling for unsaturated soils

Wenhua Wu

a

, Xikui Li

a,*

, R. Charlier

b

, F. Collin

b

a_{The State key Laboratory for Structural Analysis of Industrial Equipment, Dalian University of Technology, Dalian 116024, PR China} b_D

epartement d’Infrastructures et de Geomecanique, Institut de Mecanique et Genie Civil, Universite de Liege 4000 Liege - Sart Tilman, Belgium Received 14 September 2001; received in revised form 15 January 2003; accepted 12 September 2003

Abstract

A thermo-hydro-mechanical (THM) constitutive model for unsaturated soils is proposed in this paper based on the existing models. The inﬂuences of temperature on the hydro-mechanical behaviour in unsaturated soils are described in the model. Par-ticularly, the thermal softening phenomenon, i.e. decreases in value of the pre-consolidation pressure and in critical value of the suction of the SI (suction increasing) curve with heating process, is quantitatively modelled by using the experimental data and previous work [J. Geotech. Eng. ASCE 116 (12) (1990) 1765; J. Geotech. Eng. ASCE 116 (12) (1990) 1778; Can. Geotech. J. 25 (1988) 807; Int. J. Numer. Anal. Meth. Geomech. 22 (1998) 549; Can. Geotech. J. 37 (2000) 607; Characterisation and thermo-hydro-mechanical behaviour of unsaturated Boom clay: an experimental study. Thesis Doctoral 1999, Technical University of Catalonia (UPC), Barcelona, Spain]. Numerical modelling of the proposed THM model has also been implemented within the ﬁnite element code LAGACOM. Finally, the comparisons of the results obtained from the numerical modelling of the constitutive be-haviour with existing experimental results show that the present model can simulate the thermo-hydro-mechanical bebe-haviour in unsaturated soils with satisfaction. Some other numerical results given in the paper also validate the versatility and applicability of the present constitutive model.

Keywords: Thermo-hydro-mechanical behaviour; Constitutive model; Numerical modelling; Unsaturated soils

1. Introduction

In recently years, the thermo-hydro-mechanical be-haviour of unsaturated soils has attracted comprehen-sive attentions in engineering practice especially in nuclear waste disposal. It has been well known for a long time that the hydro-mechanical characteristics of porous media are much effected by the variation of temperature. Many efforts have been devoted to make deep understanding of the behaviour in different aspects. One of the important aspects for quantitative study of coupled, transient THM behaviour in porous media is to establish the THM mathematical model. It composed of to analyze the mechanisms governing the THM behav-iour and its evolution, to formulate the partial differ-ential governing equations and to discretise the

governing equation system for numerical solutions of the initial-boundary value problems in hands. Lewis et al. [7–9] developed the finite element model and its solution procedure for heat transfer coupled with non-linear hydro-mechanical process in saturated and un-saturated porous media. Thomas et al. [10–13] presented a fully coupled THM mathematical model based on the experimental results and previous work they achieved. They devoted much efforts on numerical modelling of coupled THM processes in porous media and its appli-cations to environmental engineering problems, partic-ularly to the numerical simulation of the THM behaviour in engineered clay barriers. Schrefler et al. [14–16] presented a general mathematical model for the analysis of THM problems in unsaturated soils with possible pollutant transport. In addition, they extended the application of the model to the numerical analysis for THM behaviour and damage phenomena of con-crete at high temperature in fire resistance engineering.

*

Corresponding author. Tel./fax: +86-411-4709186. E-mail address:[email protected](X. Li).

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Furthermore, there are some other research groups de-voting their eﬀorts in mathematical modelling. Among them are Delage et al., Alonso and Gens et al., Pande et al., Rajapakse et al.

Another important aspect in study of the THM be-haviour is to establish the constitutive model describing the coupled behaviour and its evolution and to develop the consistent algorithm for its implementation in the framework of thermodynamics combined with the in-ternal state variable theory. Because of complexities of the THM behaviours in the porous media, it brings many diﬃculties for the establishment of THM consti-tutive model. The present paper will focus on the latter aspect for unsaturated soils.

In the experimental study, Gray [17] performed iso-thermal oedometer tests at diﬀerent temperatures and revealed the dependence of the compressibility and the pre-consolidation pressure of unsaturated soils on tem-perature. Campanella and Mitchell [18] proceeded a detailed study on the thermal dilation of the soil con-stituents. Habibagahi [19] fulﬁlled isothermal consoli-dation tests at various temperatures and concluded that the permeability increases with increasing temperature due to the decrease in water viscosity. Numerous ex-periments indicated that the volumetric strain due to heating in soil samples under the drained condition de-pends on the over-consolidation ratio [20–22]. As tem-perature increases, the normal consolidated soil exhibits an irreversible contraction, which is opposite to the re-versible expansion in the over-consolidated state. Hueckel and Borsetto [1] presented an expression to describe this behaviour. Morin and Silva [23], Mitchell [24] and Towhata et al. [25] tried to explain the thermal soften phenomenon from view of the double-layer theory [23] relating to temperature variation. When temperature increases, the thickness of the bound water layer reduces as water molecules are expelled, and clay particles can get closer one to another, resulting in an irreversible volume decrease. Delage et al. [26] investigated the thermal consolidation of Boom clay. As the results, they obtained the relation between water permeability and temperature in porous media and reproduced the thermal consoli-dation behaviour in laboratory tests.

The experimental study on the constitutive relation-ship to quantitatively describe the thermo-hydro-mechanical behaviour is another the most important aspects. In 1990, Hueckel et al. [1,2] proposed an ex-tension of the modified Cam-Clay model for saturated soils to the model including the effects of temperature. Cui et al. [5] proposed a thermo-mechanical model for saturated clays based on the Cam-Clay model. Alonso et al. [27] Alonso and Gens [28] made great contribu-tions on the experimental study for constitutive model-ling of unsaturated soils. They extended the modified Cam-Clay model to integrate the effects of suction into the model and to present the well-known Alonso–Gens

model for unsaturated soils. In recent years, they ex-tended their work to include the coupled eﬀect of tem-perature with hydro-mechanical behaviour into the model. Many experimental results used in this paper are collected from the doctoral thesis [6] working at the Alonso–Gens research group.

But there have been few work carried out for the development of the THM constitutive model for un-saturated soils. Gens [29] presented a constitutive model for unsaturated soils based on a generalisation of the Cam-Clay yield surface in the stress, suction and tem-perature space. A prospective version of the LTVP model for unsaturated soils is proposed by Modaressi and Modaressi [30]. Due to lack of relevant experi-mental results, the eﬀects of temperature on suction and its related parameters were neglected in his model. In addition, the thermal yielding phenomenon described by critical thermal yield curve is also not taken into ac-count. In this paper, a new fully thermo-hydro-mechanical constitutive model for unsaturated soils is proposed as an extension of the thermo-hydro-mechanical constitutive models previously presented by Hueckel et al. [1,2] and Cui et al. [5] for saturated soils to unsaturated ones. The model is established on the basis of the experimental results [1–6] for diﬀerent types of soils, with particular attention given to volumetric strain change and the preconsolidation pressure variation with respect to temperature. The coupling behaviour between suction and temperature is also considered as one of important aspects in the model. As the present model consists of quintuple yield surfaces, the particular at-tention should be paid to the determination of the plastic strain increments. A consistent integration algo-rithm for the model with non-smooth quintuple yield surfaces, which is composed of the backward Euler re-turn mapping algorithm for integration of the rate constitutive equations and the consistent elasto-plastic tangent modulus matrix, is developed. An iterative in-tegration procedure with determination and readjust-ment of the active yield surfaces on the basis of the Kuhn–Tucker conditions is derived. The derived algo-rithm for constitutive modelling has been implemented in the FE code LAGACOM. The numerical results are illustrated in the last section to verify the presented model through the comparisons with the experimental results. Some results are further given to demonstrate capability and performance of the model.

2. Principal experimental results concerning thermo-hydro-mechanical behaviours

2.1. The effects of temperature on hydraulic properties The water in the porous medium can be classified into two different types depending on the dimension of pore

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space and its interaction with the soil matrix: inter-aggregate water (bulk water or free water which can flow in the normal condition) and intra-aggregate water (weakly bonded diffuse-layer water and strongly bonded crystal water). The inter-aggregate water is distinguished from the intra-aggregate water mainly according to the pore water velocity. Namely, adsorbed water cannot flow under normal thermal condition, whereas the bulk water is mobile due to water pressure gradients in space. However, part of the adsorbed water will be converted to the bulk water with the development of temperature.

2.1.1. The eﬀects of temperature on the retention curves As shown by the experimental results, the retention curves shift as a whole towards the left side in the suc-tion–saturation plot under increasing temperatures [6]. It implies that the suction decreases with increasing temperature under constant degree of saturation. The sensitivity of the suction with temperature change at certain constant value of the water content w can be expressed by osðwÞ oT ¼ sðwÞ a1þ b1T ; ð1Þ

where T is temperature, s suction. a1 and b1 are

empir-ical functions depending on w (or simply the coeﬃ-cients). An explicit form to predict the suction development can then be obtained as below

sðw; T Þ sðw; TrÞ ¼ a1ðwÞ þ b1ðwÞT a1ðwÞ þ b1ðwÞTr b1ðwÞ ¼ a1þ b1T a1þ b1Tr b1 ; ð2Þ where Tr is the reference temperature. It is noted that

temperature is not the unique factor aﬀecting the suction variation, especially at the high suction state. Further research work is required to improve the water retention curves. In combination of Eq. (2) with the retention curve proposed by Fredlund and Xiang [31], a new re-tention curve between Sr;w and s under given

tempera-ture can ﬁnally be obtained as below [6]

Sr;w ¼ CðsÞ 1 1þ ðaTsÞn mr ; aT¼ a a1þ b1Tr a1þ b1T b1 ; ð3Þ where Sr;w the degree of saturation. a, mr and n are the

parameters relating to the air entry value of the soil, the residual water content and the slope of the suction– saturation curve at the air entry value of the soil, re-spectively. CðsÞ is the parameter relating to suction.

2.1.2. Water permeability determination

In soil mechanics, it is usual to use the permeability to describe the hydraulic characteristics. Generally, the permeability in saturated soils keeps constant under the

room temperature. The permeability in unsaturated soils is usually determined by measuring the inflow flux and the outflow flux for a soil column sample in the oe-dometer test under the suction controlled condition with constant vertical stress or constant volumetric strain states. The effect of temperature on the relation between the permeability and the degree of saturation is signifi-cant particularly at the near saturated states. But if the degree of saturation is less than 70%, no obvious tem-perature effect on the relation appears. It is also shown that the permeability increases with increasing void ratio in constant temperature and the degree of saturation.

The water permeability in unsaturated soils can be expressed as a function of void ratio e, water content w and temperature T . It is usually assumed that the eﬀects of e; w; T on the water permeability kwcan be split and

the expression for kw is generally written in the form

kwðe; w; T Þ ¼ kw0ðeÞkwrðw; T Þ ¼ kw0ðeÞkwsðwÞkwTðT Þ; ð4Þ

where kw0ðeÞ is the intrinsic permeability depending on

the pore structure and the void ratio, kwrðw; T Þ the

rel-ative permeability, which is the function of the state variables w and T with omission of the eﬀect of water pressure. Hueckel [32] indicated that the intrinsic per-meability kw0 in saturated clay with high density is

re-lated to the eﬀective porosity n2, deﬁned as the porosity

occupied by the bulk water, and can be expressed in the form kw0¼ 1 ps2_S2 sm ðn2Þ 3 ð1 n2Þ2 ; ð5Þ

where Ssm is the speciﬁc surface area relative to the

mobile water space, p a shape factor, s tortuosity factor. A formula for kwproposed by Romero and Gens [6] for

unsaturated soils is given in the form

kwðe; w; T Þ ¼ kw0ð þ l1 TðT T0ÞÞ10ae w wref e Gs wref k ; ð6Þ

where wref is reference water content, e=Gs the saturated

water content at the actual void ratio e, lT an empirical

coeﬃcient ﬁtting the relative viscosity.

2.2. General aspects of the thermo-hydro-mechanical behaviours on oedometer tests

Usually unsaturated soils can be classiﬁed into two types according to their behaviour in volumetric stabil-ity. They are stable structured soils characterised by low porosity and swelling tendency due to wetting and drained heating and meta-stable structured soils char-acterised by high porosity and collapse tendency due to wetting and drained heating. The constitutive behav-iours observed in the experiments on the low-porosity and the high-porosity packings of unsaturated soils are separately discussed below.

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2.2.1. The results for suction controlled swelling/collapse/ shrinkage behaviours on the low-porosity packing

The experiments are fulﬁlled under constant load in isothermal and non-isothermal paths. It is observed that the irreversible strains appear in main wetting and main drying paths. The plastic strains generated in the wetting and the drying paths are mainly caused by the suction decrease (SD) yield and the suction increase (SI) yield, respectively. SI yield locus is sensitive to temperature. The suction yield value decreases with increasing tem-perature. An expression for SI yield locus was proposed as [6] SI : s bs0þ hSeik1 sIðepv;DTÞ s0 c ¼ 0; ð7Þ where hSei designates the Macauly brackets applied to the eﬀective bulk water saturation

Se¼Sr Sres 1 Sres

; ð8Þ

k1 is an empirical constant and DT stands for the tem-perature diﬀerence with respect to a reference tempera-ture Tr, s0the maximum suction ever experienced past by

the soil in Tr, sI the value of the current suction, epv the

plastic volumetric strain, Sres the residual saturation.

Fig. 1 [6] shows remarkable eﬀect of the suction on the T –ev curves in drained heating–cooling–reheating

cycles under constant net vertical stress. A yield point is easily located on the T –ev curve with the higher suction

value (i.e. s¼ 0:20 MPa) symbolised by the dashed broken line. The large irreversible strain emerges as the heating process along the line is developed to reach and beyond the critical point at which the temperature equals to 40 °C. Gens [33] has pointed out that this behaviour is associated with SD yield locus. It is ob-served from the T –ev curve with the lower value of the

suction (i.e. s¼ 0:06 MPa) symbolised by the solid line that the SD yield is already triggered at 22 °C. The cooling and the re-heating paths are characterised as the elastic process.

2.2.2. The results for suction controlled collapse(swell-ing)/shrinkage behaviours on the high-porosity packing

The experiments are also carried out under constant load in isothermal and non-isothermal paths on high porosity packing. The irreversible plastic strains gener-ated in the main wetting path and the following drying path can be clearly observed. The comparisons of vol-umetric strain–suction curves corresponding to diﬀerent net vertical stresses in two isothermal conditions are il-lustrated in Fig. 2 [6]. It is observed that the suction variations at lower level of the net vertical stresses will cause swelling strains. On the contrary, shrinkage ten-dency is observed at higher level of the net vertical stresses.

2.2.3. The volumetric strains on loading–unloading paths under controlled suction

The volumetric strains generated on loading–un-loading paths with constant suction behave in a similar tendency as that generated on the shrinkage–swelling paths with constant vertical stress. The pre-yield com-pressibility parameter j increases steadily until attaining a nearly constant value in the post-yield range. It also varies with suction as shown in Fig. 3 [6]. However, in most circumstances, the pre-yield compressibility pa-rameter j is regarded as constant approximately. The post-yield compressibility parameter kðs; T Þ increases with increasing temperature.

2.3. The results from isotropic swelling/collapse–shrink-age experiments

Isotropic swelling/collapse–shrinkage experiments are performed in constant net mean stress in order to obtain such additional information as shear strain and the strain ratio between lateral strain and axial strain for modelling of thermo-hydro-mechanical behaviour. It is observed in normal consolidation states that the main heating will cause the irreversible plastic strain. On the other hand, the quasi-reversible expansion strain due to the main heating will emerge when the soil sample is in the over-consolidation state. Two processes take place during the drained heating stages. The ﬁrst is the quasi-reversible process due to the expansion of the

mineral-Fig. 1. Drained heating–cooling cycles under constant matrix suctions and constant net vertical stress (rv ua¼ 0:026 MPa) (after [6] by Romero and Gens, 1999).

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ogical components and adsorbed water related to intra-aggregate water form. The reversible volumetric thermal strain presented [6] can be given in the form

dee v dT _p0 ¼ amþ a0ðp0Þ þ a1ðp0ÞDT ; ð9Þ

where p0 _{is constant inter-granular stress, a}

m is solid

mineral parameter which is assumed not to depend on stress level, a0 and a1 are the thermal expansion

coeﬃ-cients depending on stress states. The second is related to the irreversible meta-stable compressive phenomenon due to rearrangement of the structure of the soil

skele-ton. The collapsing strains on the SD yield locus are associated with the reduction of elastic domain. The eﬀect of temperature on the yield is reﬂected through the pre-consolidation stress p0ðs; p0ðepv;DTÞÞ and cohesion

psðs; DT Þ in the expression f ðp; q; s; p0ðp0Þ; psÞ ¼ 0 for the

yield locus.

3. The hydro-thermo-mechanical model for unsaturated soils

As discussed above, we can clearly notice that the hydro-mechanical behaviour in unsaturated soils strongly depends on temperature variation. Therefore, it is necessary to extend the hydro-mechanical constitutive models for unsaturated soils to the mechanical ones. In this paper, a new thermo-hydro-mechanical constitutive model for unsaturated soils is presented based on the existing, widespread used model – CAP model.

The general framework of the CAP model [34,35] for unsaturated soils is composed of the following four yield loci:

(1) The state boundary surface (SBS) proposed by Alonso et al. [27]

f1¼ II^r2þ m 2_ðI

r 3psðsÞÞðIrþ 3p0ðsÞÞ ¼ 0: ð10Þ

(2) The critical state line (CSL) presented by Drucker and Prager [36]

f2¼ II^rþ mðIr 3psðsÞÞ ¼ 0: ð11Þ

Fig. 2. Comparison of volumetric strains for the high-porosity packing in wetting–drying cycles at diﬀerent net vertical stresses and temperatures (after [6] by Romero and Gens, 1999).

Fig. 3. k and j at diﬀerent matrix suctions and temperatures (after [6] by Romero and Gens, 1999).

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(3) The traction model (TM) [33]

f3¼ Ir 3rt¼ 0: ð12Þ

(4) Suction increase (SI) curve also proposed by Alonso et al. [27]

f4¼ s s0¼ 0; ð13Þ

where Ir and II^r are the ﬁrst stress tensor invariant and

the second deviatoric stress tensor invariant, respec-tively, psðsÞ the cohesion, p0ðsÞ the preconsolidation

pressure and m the parameter defined by the following expression: m¼ 2 sin /c ffiffiffi 3 p ð3 sin /cÞ ; ð14Þ

/cis the frictional angle for stress paths in compression.

The pre-consolidation pressure p0ðsÞ varies with the

suction according to the loading collapse (LC) curve as follows: p0ðsÞ ¼ pc p 0 pc kð0Þj kðsÞj in which kðsÞ ¼ kð0Þðð1 cÞ expðbsÞ þ cÞ; ð15Þ where p

0 is the pre-consolidation pressure in saturated

state, pc the reference pressure, b; c are the parameter

determined by corresponding experiments. The cohesion ps also depends on the suction in the form

psðsÞ ¼ psð0Þ þ k s; ð16Þ

where psð0Þ is the initial cohesion, k is the coeﬃcient, rt

is the given traction stress. The associated plasticity is considered for this yield surface and no hardening rule is included. The CAP model is illustrated in theðIr; IIr^; sÞ

space shown in Fig. 4. Below the eﬀects of temperature on hydro-mechanical parameters in the CAP model are discussed in detail.

3.1. The thermal softening curve and the loading–collapse curve

The preconsolidation pressure p₀ in saturated state varies with temperature following the thermal

soften-ning curve (TS) proposed by Hueckel and Borsetto [1] in the form p₀ðep v;DTÞ ¼ p 0ðe p vÞ þ AðDT Þ; ð17Þ where AðDT Þ ¼ a1DTþ a2DTjDT j; ð18Þ

a1; a2 are the coeﬃcients depending on thermal

sensi-bility of the soil. It is assumed that the thermal softening is uncoupled with strain hardening and is illustrated in terms of the TS curve in p

0–T plane shown in Fig. 5 [22].

The relationship of preconsolidation pressure p0ðs; T Þ

with the suction s and temperature T in unsaturated soils is still described in format of loading–collapse (LC) curve deﬁned as p0ðs; T Þ ¼ pc p₀ðep v; TÞ pc ½kð0Þj=½kðsÞj : ð19Þ

However, experimental results indicate that the plas-tic slope kðsÞ varies with the heating–cooling cycles, namely, k¼ kðs; T Þ. A simpliﬁed expression for k ¼ kðs; T Þ used in the present work is given in an uncoupled form as below

kðs; DT Þ ¼ kðs; TrÞ þ b1DTþ b2DTjDT j

¼ kð0; TrÞ½ð1 cÞ expðbsÞ þ c þ b1DT

þ b2DTjDT j; ð20Þ

where b1;b2 are the empirical coeﬃcients. Fig. 6

illus-trates the LC curves corresponding two diﬀerent tem-peratures.

The cohesion also varies with temperature and can be expressed in the form

psðs; DT Þ ¼ psð0; TrÞ þ kðs; DT Þs: ð21Þ

However, it is found according to the experimental ob-servations that the variation of the cohesion with in-creasing temperature is much smaller in value than that relating with increasing suction. For the sake of conve-nience in the derivation of the relative formulations, it is assumed that the cohesion is only related to the suction.

Fig. 4. The CAP model in theðIr; II^r; sÞ space.

0 10 20 30 40 50 60 0 204 0 608 0

The experimental data The data points on the fitting curve The fitting curve

T(˚C)

p

0 *_(MPa)

Fig. 5. Thermal soften function in p0–T plane (after [22] by Eriksson, 1989).

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Temperature is supposed to have no inﬂuence on the traction stress. SI curve also varies with increasing temperature as a lower air-entry value under surface-tension reasoning is expected at higher temperatures. SI curve can then be chosen [6] as that shown in Eq. (7). 3.2. The thermal yield curve

In the following the irreversible deformation due to the eﬀect of temperature variation is discussed. The ex-perimental results indicate that increasing temperature beyond a certain prescribed value T0 will cause plastic

strains even in over-consolidation states. This plastic yielding phenomenon is simply described by the so-called thermal yield curve (TYC). It should be pointed out that it is rather cursory to use a straight line (T ¼ T0)

for simulating the thermal yield. A more sophisticated form of the thermal yield curve needs to be built on the basis of the related experimental results. Cui et al. [5] proposed a form for TYC for saturated soils, as shown in Fig. 7, given by

T ¼ ðTr T0Þ expðb3p0Þ þ T0; ð22Þ

where b3 is the parameter deﬁning the curvature of the

curves. However, the above form of TYC is not included the eﬀects of suction. Further research work is required to improve the thermal yield curve.

3.3. The yield surfaces

To take into account the eﬀects of temperature, the multiplicate yield surfaces of the CAP model expressed by Eqs. (10)–(13) and (22) are extended and given in the following yield formulae:

f1¼ II^r2þ m 2_ðI r 3psðsÞÞðIrþ 3p0ðs; DT ÞÞ ¼ 0; f2¼ II^rþ mðIr 3psðsÞÞ ¼ 0; f3¼ Ir 3rt¼ 0; f4¼ s bs0þ hSei k1 ðsIðepv;DTÞ s0Þc ¼ 0; f5¼ T T0¼ 0 or f5¼ T ðTr T0Þ expðb3p0Þ T0: ð23Þ As the four independent variables ðIr; II^r; s; TÞ are

in-cluded in the model, it is impossible to illustrate it in only one ﬁgure. We give three plots in the following three diﬀerent 3D spaces as shown in Fig. 8.

Fig. 7. Thermal yield curve in p0–T plane.

Fig. 8. The constitutive model in (a) theðIr; II^r; sÞ space with diﬀerent temperature T2> T1; (b) theðIr; IIr^; TÞ space; (c) the ðIr; s; TÞ space. Fig. 6. Loading–collapse curves in p0–s plane in two diﬀerent

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3.4. The constitutive relations in rate form

Let us express the present constitutive model in the rate form. The total strain rate is additatively decom-posed of three parts owing to the mechanical, thermal and suction eﬀects, respectively, and each part of them can be further additatively decomposed into elastic and plastic portions, i.e.

_eij¼ _emij þ _e s ijþ _e T ij ¼ _em;e ij þ _e m;p ij þ _e s;e ij þ _e s;p ij þ _e T;e ij þ _e T;p ij : ð24Þ

The total stress rate can be expressed by the HookÕs law in terms of the elastic portion of the mechanical strain rate as follows: _ rij¼ Cijkle _e m;e kl ; ð25Þ where Ce

ijkl is the elastic compliance tensor. The elastic

strain _es;e_ij due to the suction variation is given by _es;e_ij ¼ js

ð1 þ eÞ _s ðs þ paÞ

dij¼ heij_s; ð26Þ

where js is elastic stiﬀness parameter for changes in

suction. From the experimental results, the elastic strain _eT;eij due to the temperature change is evaluated through

a thermal expansion parameter a2

_eT;eij ¼ a2T_dij¼ Tije _T : ð27Þ

The plastic strain rate _em;pij due to mechanical eﬀect is

given according to the plastic ﬂow rule _em;p_ij ¼ _kp og

orij

ð28Þ in which g is the plastic potential function and the non-associated plasticity is considered. one can obtain its associated form by simply assuming g¼ f , f is the as-sociated yield function.

The plastic strain part corresponding to the suction variation is written as _es;p_ij ¼ks js ð1 þ eÞ _s ðs þ paÞ dij¼ h p ij_s; ð29Þ

where ks is plastic stiﬀness parameter for changes in

suction.

The plastic strain part due to thermal eﬀect can be determined according to the following consistent yield condition: of oTdTþ of oeT;pij deT;pij ¼ 0 ð30Þ

from which it is given that

_eT;pij ¼ T p

ijT :_ ð31Þ

As CSL is activated by temperature variation, Tijpcan be

calculated as below T_ijp ¼de T;p ij dT ¼ of op0 op0 op 0 op 0 oT of op0 op0 op 0 op 0 oeT;p_ij ¼ op 0 oT op 0 oeT;p_ij ¼ ðkð0Þ jÞða1þ 2a2DTÞ ð1 þ eÞp 0 : ð32Þ

While TYC is activated, the volumetric plastic strain can be computed according to the expression proposed by Cui and Delage as

_eT;p_ij ¼ expðapT_Þ aapT_ 1; ð33Þ

where a is the parameter equal to one approximately, ap

can be determined by corresponding experiments. Substitution of Eqs. (25)–(29) and (31) into Eq. (24) gives _eij ¼ Ce 1 ijklr_klþ _k p og orij þ he ij_sþ h p ij_sþ T e ijT_ þ T p ijT :_ ð34Þ

For a yield function f with the hardening/softening rule depending on the internal state variable h, the plastic consistency condition of the yield function can be writ-ten as _f ¼of orr_þ of oh_h þ of os_sþ of oT _ T ¼ 0; ð35Þ where _ rij¼ Ceijklb_ekl _e m;p kl _e s;e kl _e s;p kl _e T;e kl _e T;p kl c; ð36Þ _h ¼ dh dep valk _kp þ vals _s þ valT _T ; ð37Þ

valk, valsand valT stand for the tensorsog=orij, h p ij and

T_ijp, respectively. The plastic volumetric strain ep

v is taken

as the internal state variable h to govern the evolution of the multiplicate yield surfaces. Substitution of expres-sions (36) and (37) into Eq. (35) gives

_f ¼ of orij Ce_ijkl _ekl _kp og orij he kl_s h p kl_s T e klT_ T p klT_ þof oh oh oep valk _k p h þ vals _s þ valT _T i þof os_sþ of oT _ T ¼ 0 ð38Þ and results in _kp_¼ of orijC e ijkl_eklorofijC e ijkl h e klþ h p kl _s of orijC e ijkl T e klþ T p kl _{_} Tþof oh oh oep vals _s þ valTT_ h i þof os_sþ of oTT_ of orijC e ijkl og orkl valk of oh oh oep : ð39Þ

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Finally, the stress rate in the consistent form can then be computed as follows: _ rij¼ Ceijkl " of orabC e abklC e ijcd og orcd of ormnC e mnop og orop valk of oh oh oep # _ekl Ce ijkl h e kl " þ hp_klþ C e ijkl og orkl of ormnC e mnop og orop valk of oh oh oep of os þ vals of oh oh oep of ormn Ce mnkl h e kl þ hp_kl # _s Ce ijkl T e kl " þ Tklp þ C e ijkl og orkl of ormnC e mnop og orop valk of oh oh oep of oT þ valT of oh oh oep of ormn C_mnkle T_kle þ Tklp #_{_} T : ð40Þ

It is remarked that in contrast with single yield surface plasticity, a particular attention to determine the plas-tic strain increments has to be given to the corner where two yield surfaces are active. A backward Euler re-turn mapping procedure with determination of the ac-tive yield conditions on the basis of the Kuhn–Tuker conditions for non-smooth multi-surface plasticity is required [35,37].

4. Numerical examples

As the ﬁrst example, a soil cube in drained condi-tion subjected to a step heating uniformly applied over the whole region of the cube is considered. The uni-formly distributed pressure loading shown in Fig. 9 is applied on the three edges of the cube. The purpose of the simulation is to verify the present constitutive

model via a comparison of the results [6] obtained by the modelling with corresponding experimental ones. The finite element mesh with only one eight-node el-ement is used. The material property data used for the element are shown in Table 1. Two cases with two different initial consolidation pressures applied on the soil cube are performed. The first case is performed in the over-consolidation state, in which rm pa¼ 0:10

MPa and s¼ pa pw¼ 0:20 MPa are speciﬁed,

tem-perature increases from 295 to 355 K. The second is performed in the normal consolidation state, in which rm pa¼ 1:0 MPa and s ¼ pa pw¼ 0:20 MPa are

Fig. 9. The soil cube for isotropic drained test.

Table 1

Parameters relating to thermo-hydro-mechanical behaviour

kð0Þ 0.057 a 1.0 j 0.005 b )0.037 b 1.336 MPa1 _k ₁ c 0.015 ks 0.25 p 0ðTrÞ 1.0 MPa js 0.11 pc 0.585 MPa a1 )1500 Pa/K /c 35° a2 63 Pa/K T0 373 K a2 0.000050 (K1) Tr 295 K m 0.4 e0 0.679 qw 1000 kg/m3 s0 18 MPa qs 2750 kg/m3 Volumetric strain 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 Temperature, T(C) Volumetric Strain (%) (a) Volumetric strain -0.6 -0.4 -0.2 -1 -0.8 -0.6 -0.4 -0.2 0 0 10 20 30 04 50 60 70 80 90 0 10 20 30 04 50 60 70 80 90 Temperature, T(C) Volumetric strain (%) Numerical simulation Experimental results Numerical simulation Experimental results (b)

Fig. 10. Drained volumetric strains versus temperatures for numerical simulation and experimental results in (a) overconsolidation state (b) normal consolidation state.

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specified with the same heating process as applied to the first case. Fig. 10 illustrates the variations of the volumetric stain versus increasing temperature in both cases. The experimental results are also illustrated in the same figure.

It is observed from the results for this example that eﬀect of the thermal softening on the yield surfaces caused by heating process is compensated by that of strain hardening so that the mechanical balance still remains. As a consequence, additional irreversible vol-umetric strains characterised as the thermal plasticity phenomenon occur and develop.

The second example considers a soil column, which is 5 m deep and 1 m wide. Fourteen non-uniformly divided eight-node element mesh shown in Fig. 11 is used for numerical modelling of the example. The initial temperatures 353 and 293 K are prescribed for the eight nodes of the ﬁrst element and all of the rest nodes in the mesh, respectively. As the boundary condition, the temperature equal to 353 K for the three nodes located at the top of the column is ﬁxed in the heat conduction process. The initial stress equal to 1.2 MPa is applied on the whole column, which is as same as the preconsolidation pressure prescribed to the column. The material parameters used in the ex-ample are listed in Table 2. The four cases

corre-sponding to different loading and critical temperature conditions are considered. The first and the second cases are carried out under the increasing vertical load path ranging from 1.2 to 2.0 MPa applied on the top of the soil column as shown in Fig. 11. The critical temperatures defined as material parameters for the two cases are different. For the first case the critical temperature is equal 363 K, hence, the heat conduc-tion process along the column in the first case will not trigger TYC causing plastic strains. Whereas the crit-ical temperature specified as the material parameter for the second case is 343 K. Therefore, the front of the zone of the column, at which the temperature exceeds the prescribed critical value and the plastic strains are induced with the heat conduction process, will be gradually advanced from the top toward the bottom of the column. The third and the forth cases concern with the constant vertical load of 1.2 MPa applied on the top of the column. The critical tem-peratures specified as the material parameters for the third and the forth cases are assumed as the same as those specified in the first and the second cases, re-spectively. The results for the vertical strain distribu-tions along the column for the four cases at a series of time steps are illustrated in Fig. 12.

Through the comparisons of the four resulting volu-metric strain distributions it can be seen how the thermal yield surface inﬂuences to the resulting volumetric strains. It indicates that TYC will play an important role in the present constitutive model.

The heating example on a sample of unsaturated Boom clay proposed as the Benchmark Test 2.2 is, as the third example, simulated. The sample was com-pacted into a cylindrical stainless steel cell with the inner diameter 15 cm, the inner height 15.6 cm and the thickness 3.6 cm. The heater with length of 10 cm and the diameter 2 cm is inserted into the centre of the cell as shown in Fig. 13. The domain taken for the modelling of this example is composed of the soil sample, the porous plate and the steel case but no heater. The ﬁnite element mesh with 629 nodes and

Fig. 11. Geometry and ﬁnite element mesh of the soil column.

Table 2

Parameters related to thermo-hydro-mechanical behaviour

kð0Þ 0.4041 a 1.0 j 0.014 b )0.037 b 0.41 MPa1 _k ₁ c 0.30 ks 0.25 p 0ðTrÞ 1.6 MPa js 0.11 pc 1.20 MPa a1 )1500 Pa/K /c 35° a2 63 Pa/K T0 373 K a2 0.000050 (K1) Tr 295 K m 0.4 e0 0.667 qw 1000 kg/m3 s0 18 MPa qs 2750 kg/m3

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179 eight-node elements are used. The initial values of the parameters are listed in Table 3. The steel case is impermeable to water. The gas pressure is assumed constant equal to atmosphere pressure during the whole process. The test is divided into the two phases. In the 1st phase, the heater temperature increases from 295 to 373 K. Correspondingly, the water

pres-sure at the zone near the porous plate also increases from )78.5 to 1.1 MPa. This phase lasts 10,000 s. During the 2nd phase (from 10,000 s to 2401.6 h) the heater temperature remains constant. Fig. 14 illus-trates the evolution of the water intake obtained by present modelling and compared with the experimental

Table 3

The initial conditions

Total stress r¼ 0 Pa

Degree of saturation Sr¼ 0:49

Estimated pore water pressure pw¼ 78:5 MPa

Gas pressure pa¼ 100 kPa

Initial temperature T¼ 293 K Water intake 0.0 0E+00 5.0 0E+01 1.0 0E+02 1.5 0E+02 2.0 0E+02 2.5 0E+02 3.0 0E+02 3.5 0E+02 4.0 0E+02 0 500 1000 1500 2000 2500 3000 Time [hour] Volume [cm3] Experimental results Numerical results

Fig. 14. Comparisons between experimental and numerical results on the water intake.

Vertical strain Vertical strain -2 .50E-02 -2 .00E-02 -1 .50E-02 -1 .00E-02 -5 .00E-03 5. 00E-03 - 6 -5 -4 -3 -2 -1 0 y Time = 60000 Time = 60000 Time = 370000 Time = 2850000 Time = 6850000 Time = 11000000 Eps y (c) Vertical strain Vertical strain -1.60E-02 -1.40E-02 -1.20E-02 -1.00E-02 -8.00E-03 -6.00E-03 -4.00E-03 -2.00E-03 2.00E -03 - 6 -5 -4 -3 - 2 - 1 0 y Time = 370000 Time = 2690000 Time = 6690000 Time = 11000000 Eps y (d) (a) (b) -7 .00E -02 -6 .00E -02 -5 .00E -02 -4 .00E -02 -3 .00E -02 -2 .00E -02 -1 .00E -02 0. 00E+00 1. 00E-02 - 6 -5 -4 -3 -2 -1 0 y Ti me = 60000 Ti me = 1030000 Ti me = 2530000 Ti me = 4030000 Ti me = 5530000 Ti me = 7030000 Ti me = 8530000 Ti me = 10050000 Eps y -6 .00E-02 -5 .00E- 02 -4 .00E- 02 -3 .00E- 02 -2 .00E- 02 -1 .00E- 02 1. 00E-02 -6 - 5 -4 -3 -2 -1 0 y Time = 60 000 Time = 75 0000 Time = 27 10000 Time = 47 10000 Time = 57 10000 Time = 73 10000 Time = 86 10000 Time = 10 050000 Eps y 0. 00E+00 0.00E+00 0.00E+00

Fig. 12. The vertical strain distributions along the column for the four cases at a series of time steps.

Fig. 13. The model diagram for thermo-hydro-mechanical cell (Benchmark Test 2.2) with an inserted heater (distances in cm).

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results. Fig. 15 illustrates the contours obtained by the numerical modelling for saturation, suction, deviatoric and mean strain/stress, plastic volumetric strain, tem-perature at 2401.6 h. The main purpose of this ex-ample is to show the capability and applicability of the present model in modelling rather complicated 2D problems.

5. Conclusion

A thermal-hydro-mechanical constitutive model for unsaturated soils is presented in this paper. The model is based on the four component yield surfaces of the CAP model and on the experimental results obtained for diﬀerent types of soils. The extension of the model to including the eﬀect of temperature is embodied through the thermal softening and the suction variation with

respect to temperature, as well as the thermal eﬀect to the hydraulic properties. The present model has been implemented in the ﬁnite-element code LAGACOM. The comparisons of the numerical simulations with experimental results show that the thermal-hydro-mechanical behaviour is properly modelled in the present work.

Acknowledgements

This work is sponsored by Ministry of Education of Belgium under the international joint research project ‘‘Qualite et durabilite de la protection des nappes aquiferes sous sites dÕenfouissement technique avec barrieres argileuses dÕetancheite’’ and The National Natural Science Foundation of China under the project

Fig. 15. Contours obtained by the numerical modelling of Benchmark Test 2.2 at 2401.6 h for: (a) saturation; (b) suction; (c) mean stress; (d) mean strain; (e) deviatoric stress; (f) deviatoric strain; (g) plastic volumetric strain; (h) temperature.

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Nos. 50278012, 10302005, 59878009 and 10225212. The support is greatly acknowledged.

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